On regular trace languages

We compare two notions of coding on traces, called coding and strong coding, in relation with the decision problem of the existence of a coding between two given trace monoids. We positively solve this problem if the first monoid is either a direct product of free monoids, or a free product of free commutative monoids. We show how the situation differs in the case of strong codings.

Trace monoids are obtained from free monoids by defining a subset I of pairs of letters that are allowed to commute. Most of the work of this theory is an attempt to relate the properties of these monoids to the properties of I. Following the work initiated by Buchi we show that when I is an equivalence relation (the trace monoid is then a free product of free commutative monoids) it is possible to define a second order logic whose models are the traces viewed as dependence graphs and which characterizes exactly the sets of traces that are rational. This logic essentially utilizes a predicate based on the ordering defined by the dependence graph and a predicate related to a restricted use of the comparison of cardinality.

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.KeywordsWord ProblemInverse SemigroupCayley GraphDependence GraphDependence RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Schemes of concurrent programs are considered. The result of a scheme is defined as a set of traces, where each trace is a partially ordered set of symbol occurrences. It is shown that to each scheme corresponds a set of equations determining the result of the scheme; it is shown how these equations can be solved and that the solutions of these equations are regular trace languages. Next, a notion of action systems is introduced; an action consists of its resources and its transformation. Some properties of action systems are shown. Interpretations of schemes are defined as mappings which assign actions to scheme symbols. Interpreted schemes can be regarded as concurrent programs. It is shown how the results of schemes can be lifted (via interpretations) to the results of programs. Some examples of applications of the described methods to prove concurrent programs correct are given.

Zielonka’s theorem shows that each regular set of Mazurkiewicz traces can be implemented as a system of synchronized processes provided with some distributed control structure called an asynchronous automaton. This paper gives a new algorithm for the synthesis of a non-deterministic asynchronous automaton from a regular Mazurkiewicz trace language. Our approach is based on an unfolding procedure that improves the complexity of Zielonka’s and Pighizzini’s techniques: Our construction is polynomial in terms of the number of states but still double-exponential in the size of the alphabet. As opposed to Métivier’s work, our algorithm does not restrict to acyclic dependence alphabets.

The word problem for inverse monoids generated by a set $Gamma$ subject to relations of the form $e=f$, where $e$ and $f$ are both idempotents in the free inverse monoid generated by $Gamma$, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node $x$ to a node $y$ that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, some results on free partially commutative inverse monoids are presented.

Easy multiplications I. The realm of Kleene's theorem

Set Constraints in Some Equational Theories

An extension of Kleene's and Ochmański's theorems to infinite traces

A totally ordered monoid—or tomonoid , for short—is a commutative semigroup with identity S equipped with a total order \les that is translation invariant, i.e. , that satisfies: \forall x, y, z∈ S, x\les y\;\Rightarrow \; x+z \les y+z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice \Z n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell.

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory.

Several authors have studied the question of when the monoid ring \(DM\) of a monoid \(M\) over a ring \(D\) is a right and/or left fir (free ideal ring), a semifir, or a \(2\)-fir (definitions recalled in §1). It is known that for \(M\) nontrivial, a necessary condition for any of these properties to hold is that \(D\) be a division ring. Under that assumption, necessary and sufficient conditions on \(M\) are known for \(DM\) to be a right or left fir, and various conditions on \(M\) have been proved necessary or sufficient for \(DM\) to be a \(2\)-fir or semifir. A sufficient condition for \(DM\) to be a semifir is that \(M\) be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid \(M\) which is not such a direct limit, but satisfies all the known necessary conditions for \(DM\) to be a semifir. It is an open question whether for this \(M,\) the rings \(DM\) are semifirs. We note here some reformulations of the known necessary conditions for a monoid ring \(DM\) to be a \(2\)-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recover Cedó’s results for both constructions. Any homomorphism from a monoid \(M\) into \(\mathbb {Z}\) induces a \(\mathbb {Z}\)-grading on \(DM,\) and we show that for the two monoids just mentioned, the rings \(DM\) are “homogeneous semifirs” with respect to all such nontrivial \(\mathbb {Z}\)-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If \(M\) is a monoid such that \(DM\) is an \(n\)-fir, and \(N\) a “well-behaved” submonoid of \(M,\) we prove some properties of the ring \(DN.\) Using these, we show that for \(M\) a monoid such that \(DM\) is a \(2\)-fir, mutual commutativity is an equivalence relation on nonidentity elements of \(M,\) and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or of infinite cyclic monoids. Several open questions are noted.

A generalization of the Schützenberger product of finite monoids

We solve the open problem of characterizing those concurrency alphabets for which the equivalence problem for regular trace languages is decidable. We also give a characterization for the cases in which the intersection emptiness problem is decidable.

Two-sided wreath product of categories